Asymptotic complements in the integers

Let W be a nonempty subset of the set of integers Z. A nonempty subset C of Z is said to be an asymptotic complement to W if W+C contains almost all the integers except a set of finite size. The set C is said to be a minimal asymptotic complement to W if C is an asymptotic complement to W, but C∖{c} is not an asymptotic complement to W for every c∈C. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in Z and deal with a problem of Nathanson on their existence and their inexistence.